Viscosity, and more generally, viscoelasticity, are properties of liquids and solids that relate the shear forces generated by or applied to a material to the amount of shear deformation or flow. While the invention applies equally well to viscoelasticity, the present discussion will be limited to viscosity measurement for simplicity. Viscosity is of widespread interest in many manufacturing environments and is measured as a primary quality of some products and as a secondary quality (a means of monitoring process state) in other processes.
Viscosity describes the force required in order to make successive molecular layers of a liquid move past each other at a given rate of shear (“shear rate”). If one considers a liquid flowing past the walls of a container, the liquid will ideally have no motion relative to the wall at the interface and will have increasingly higher velocities as one observes points successively further from the wall. The shear rate is defined as the gradient of the velocity of the liquid parallel to the surface (meters per second) with increasing distance from the surface (meters). The units of shear rate are 1/seconds. The shear stress is the amount of force per unit area that must be applied in order to cause the motion. While the fluid may have a characteristic flow (and thus a characteristic shear rate) or may be stationary, all measurements of viscosity to date are based on the measurement of shear stress vs. shear rate under an imposed motion of the fluid. Throughout this disclosure, “shear rate at which the viscosity of a fluid” should be taken to mean the shear rate at which the viscosity of the fluid is measured, which may differ substantially from the characteristic shear rate of the fluid in its intended application or point of measurement.
Intrinsic viscosity is defined as the ratio of the shear stress to the shear rate and has units of pressure (force per unit area) times seconds, or Pascal-seconds. Intrinsic viscosity is typically measured using a rotating cylinder of known diameter at a controlled rate of rotation concentric within another stationary cylinder. Knowledge of the torque on the rotating cylinder and its rate of rotation, along with geometrical factors, allows one to measure the intrinsic viscosity, or simply the “viscosity”, η.
While this method is widely used and highly accurate, it is difficult to successfully apply to process control or to operate on moving platforms. The ability to accurately measure nonlinearities of viscosity as a function of rotational rate notwithstanding, the use of moving parts limits utility and reliability. The requisite sample sizes and difficulty in setting up and cleaning up a measurement motivate other alternatives.
Hydrodynamic properties of liquids are quantified by the speed of shear sound waves in a liquid. An ideal liquid, having neither shear elasticity nor viscosity, cannot support a shear sound wave. An elastic solid having a shear stiffness, μ, can support a shear acoustic wave that propagates through the solid much as the better known compressional sound wave. Viscous liquids can support a shear sound wave; however the wave decays as it travels and is only able to travel a few wavelengths before being totally damped out by frictional losses. Nonetheless, these “sound waves” are related to the flow of liquids in confined geometries, such as capillaries, pipes and the spaces between moving parts in machinery. These flows are governed by the ratio of the intrinsic viscosity, η, to the density of the material, ρ. The ratio is known as the “kinematic viscosity”, ηk=η/ρ, and has the units of area over time (m2/s).
Kinematic viscosity is typically measured using a controlled glass capillary tube maintained in a water bath with a reproducible force (gravity, air pressure, or vacuum) applied. The velocity of a sample through the capillary is measured. The “shear rate” is a function of the speed of the sample and the geometry of the capillary. While this method enjoys widespread use and there exist numerous clever methods of automating sample injection and measurement, the approach is not without its problems and the industry is in need of alternative solutions. A principal problem with this method is the discontinuous and sampled nature of the measurement, in which it is typically performed off-line in a laboratory separate from the process being monitored and controlled. Other drawbacks are the size of the apparatus and its susceptibility to vibration and orientation of the capillary.
Acoustic waves offer an attractive method of measuring the viscoelastic properties of liquids and solids, including oils, paints, inks, polymers and glasses. Prior art has indicated a number of methods by which viscosity is measured, as well as methods in which a change in viscosity is used as an indication of a chemical measurement. While the invention is equally applicable to either scenario, the disclosure will focus on the measurement of viscosity itself.
The propagation of shear acoustic waves in a liquid, while difficult to instrument reliably, offers a direct measurement of the kinematic viscosity. The square of the phase velocity of the wave in the liquid divided by the radian frequency of the wave, ω, is a direct measure of ηk. The method is not employed due to the substantial variations in the size of a wavelength with changes in viscosity and the significant signal to noise ratio issues associated with measuring the phase velocity of a heavily damped wave over a small number of acoustic wavelengths.
The compressional wave amplitude and velocity are a weak function of the shear viscosity. Assuming that the compressional stiffness constant, λ, has no loss of its own (elastically compressible liquid), the effective elastic modulus of the compressional plane wave is λ+2jωη, where j is the complex operator and ω is the radian frequency. Assuming ωη<<λ, the viscoelasticity is a weak and approximately linear function of the viscosity. The square of the compressional wave velocity may be expressed as the ratio of the effective elastic modulus to the density. Measurements of the compressional wave velocity are often correlated to the viscosity by assuming that the compressional modulus, λ, and the density, ρ, are unchanged. These quantities are not typically invariant and therefore the results of measurements described in the like of U.S. Pat. No. 6,439,034 to Farone et al. are inaccurate; however this method is applied to process control as an approximation. At present there is no meaningful definition or measurement of shear rate in this method and the method is typically recommended only for Newtonian liquids, that is liquids in which the viscosity is independent of the shear rate and is constant. The method suffers constraints of fixed geometry between the transducers that sometimes does not fit the flow requirements within a process, and as the Farone patent discloses only measurement of the phase difference between the longitudinal and shear wave without consideration of the amplitude or attenuation of the wave, it suffers from operating under certain meaningful, but unknown operating conditions, such as the signal amplitude, which are known to change the measured quantity. This limitation is specifically important in liquids that exhibit non-linear viscosity response to shear rate.
In addition to the propagation of acoustic waves through a material, it is possible to employ acoustic waves in an adjacent solid to measure the power transfer into the viscous liquid. Power transfer from one medium to the other is governed by the ratio of the acoustic impedances of the materials and is well known to one skilled in the art. Power transfer of acoustic energy between a solid waveguide and an adjacent liquid forms the basis of several viscometers. The rate of energy transfer (power loss) is dependent on the relative acoustic impedances of the waveguide material and the adjacent liquid in a manner well-known to one skilled in the art. The acoustic impedance of the shear wave in the solid waveguide is the square root of the product of the density, ρ, and the shear elastic stiffness, μ. It is predominantly real (resistive) and is analogous to a nearly-lossless transmission line's characteristic impedance in electromagnetics. The acoustic impedance of a shear wave in a viscous liquid is the square root of the product of the stiffness term, jωη, with the density, ρ. The characteristic impedance of the viscous liquid is typically very small compared to the elastic solid, resulting in a low power transfer. At low viscosity the power transfer is proportional to (ωρη)1/2.
One method of performing this measurement is to immerse a resonator, typically a disc of quartz crystal of AT cut, into the liquid and to measure the shift in resonant frequency or loss at resonance. This method has been plagued by poor reproducibility when used with affordable instrumentation, primarily due to the lack of a differential measurement.
In all of the acoustic wave methods, it has generally been virtually impossible to correlate the “shear rate” at which the measurement occurs to the “shear rate” of such reference methods as the rotating spindle or the capillary tube. Therefore, while the acoustic wave methods are structurally superior, having no moving parts and offering continuous in situ measurement, they are not widely accepted due to the unknown conditions under which the measurements occur.
Ideally the viscosity of a liquid is a constant property that is independent of the rate of shear. That is, the stress is a linear function of the shear rate and the slope of the stress vs. shear rate, or “viscosity”, measured by one instrument type or at one shear rate would be identical to that measured by another instrument type or at another shear rate. This is, unfortunately not the case, and all liquids exhibit nonlinearity beyond a given stress limit or shear rate limit. There are a wide variety of mechanisms for the observed nonlinearities and a correspondingly large number of models to describe the effect.
A common model for liquid is known as the Maxwellian model, in which the liquid is assumed to have a fixed stiffness in parallel with viscosity, so that the Maxwellian liquid behaves as a semi solid beyond a given shear rate. This model is widely adapted in the field of acoustic sensors because it quickly lends itself to the analysis of harmonic motion such as in waves. In the Maxwellian model it is assumed that the stiffness constant, μ, and the viscosity, η, are both constant and that the properties of the material are linear with changes in the amplitude of the acoustic wave, varying only with changes in the frequency of the motion. While many liquids exhibit behaviors consistent with this model at relatively low frequencies and amplitudes, this has been found to be an erroneous assumption in general. The most significant error to result from this assumption is to equate the shear rate of the sensor with the frequency of the harmonic motion. Thus, one sees sensors based on 10 MHz quartz crystals described as 107 1/seconds shear rate viscometers, when in fact, they exhibit unknown and uncontrolled shear rates on the order of 102 to 105 1/seconds, depending on the acoustic amplitude and other geometric factors.
The preferred embodiment of the present invention has been analyzed and was found to have an uncontrolled shear rate of between 103 and 2×107 1/seconds. The shear rate is a function of several factors, most notably the power level of the acoustic wave transmitted into the sensor and the viscosity of the liquid being measured.
In U.S. Pat. No. 3,903,732 to Rork, a sensor is provided in which a transducer therein provides an energy transduction from one form of energy to another and in doing so has surfaces of the transducer placed in motion in directions such that shear wave energy is imparted to those portions of the fluid of interest adjacent to those surfaces. The transducer has electrical input terminals to which oscillator circuitry is attached whereby the circuit oscillation frequency is a resonant frequency associated with the moving surfaces when submerged in a fluid. Rork acknowledge a dependence of a material's viscosity on the magnitude of the applied stress, and further states that “This sensor measures the point fluid viscosity of non-Newtonian fluids for the particular shear stress applied.” Rork then discuss a means to potentially control the stress. The predominant practice in the fields of viscometry and rheology is to measure viscosity as a function of shear rate and not of stress. This practice is addressed by the present invention which, in its various aspects, measures, controls and varies the shear rate of the sensor as opposed to the peak stress of the transducer. Furthermore, the present invention allows a correlation of the shear rate to mechanical viscometers such as the Brookfield™ (Brookfield Inc. Middleboro, Mass. U.S.A. ) viscometer which is considered the standard measurement device for this purpose.
Therefore a promising class of sensors for measuring viscosity or for measuring chemical processes that induce a change in viscosity are underutilized by the industry, primarily because those sensors do not provide for measurement or control of the shear rate at which the viscosity is being measured. It is further found that this class of sensors are being underutilized because it is not presently possible to maintain a constant shear rate in the sensor over a range of viscosities or to reproducibly alter the shear rate of the measurement over a desired range in a given sample. It is an object of the present invention to overcome various aspects of the limitations described above. It is a further object of the invention to provide for measurement of viscosity at a controlled shear rate, preferably representative of the intended application.